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Notes on generalised spin structures

Andrew D. K. Beckett

TL;DR

The paper develops a comprehensive framework for generalised spin structures by extending spin geometry to Spin$^G$-type groups $Spin^G(s,t)=(Spin(s,t)\times G)/\mathbb{Z}_2$, enabling twisted spinors and broader geometric constructions. It introduces a covariant Lie derivative and a covariant Cartan calculus on associated bundles, and defines a symmetry algebra $\mathfrak{symm}(\widehat{\varpi},\mathscr{A})$ that properly represents Killing symmetries in the twisted setting via an extension by the gauge sector with curvature $F$. The work then treats homogeneous generalised spin structures, establishing a classification in terms of $H$-conjugacy classes of lifts of the isotropy representation to $Spin^H(V)$ and applying Wang’s/Nomizu framework to invariant connections and curvatures. Collectively, the results provide a robust toolkit for analyzing geometric and physical applications where spinors are twisted by a gauge group, including reducibility criteria, time orientation, and homogeneous space constructions, with implications for Dirac-type operators and gauge-coupled spin geometry.

Abstract

We review some definitions and basic notions relating to generalised spin structures and introduce the notion of reducibility. We discuss connections on these structures, define a covariant Lie derivative for associated bundles and develop a covariant Cartan calculus. We introduce an extension of the Lie algebra of Killing vectors, the symmetry algebra, and show that it has a representation on sections of associated bundles. We discuss homogeneous generalised spin structures and provide a characterisation of them in terms of lifts of the isotropy representation.

Notes on generalised spin structures

TL;DR

The paper develops a comprehensive framework for generalised spin structures by extending spin geometry to Spin-type groups , enabling twisted spinors and broader geometric constructions. It introduces a covariant Lie derivative and a covariant Cartan calculus on associated bundles, and defines a symmetry algebra that properly represents Killing symmetries in the twisted setting via an extension by the gauge sector with curvature . The work then treats homogeneous generalised spin structures, establishing a classification in terms of -conjugacy classes of lifts of the isotropy representation to and applying Wang’s/Nomizu framework to invariant connections and curvatures. Collectively, the results provide a robust toolkit for analyzing geometric and physical applications where spinors are twisted by a gauge group, including reducibility criteria, time orientation, and homogeneous space constructions, with implications for Dirac-type operators and gauge-coupled spin geometry.

Abstract

We review some definitions and basic notions relating to generalised spin structures and introduce the notion of reducibility. We discuss connections on these structures, define a covariant Lie derivative for associated bundles and develop a covariant Cartan calculus. We introduce an extension of the Lie algebra of Killing vectors, the symmetry algebra, and show that it has a representation on sections of associated bundles. We discuss homogeneous generalised spin structures and provide a characterisation of them in terms of lifts of the isotropy representation.

Paper Structure

This paper contains 20 sections, 10 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. A note on this note
  3. Organisation
  4. Basic notions
  5. The spin-$G$ group
  6. Spin-$G$ structures
  7. Reducibility
  8. Time orientability
  9. Representations of the spin-$G$ group
  10. Associated bundles and sections
  11. Connections and curvatures
  12. Covariant Lie derivative and Cartan calculus
  13. Covariant Lie derivative
  14. Covariant Cartan calculus
  15. The symmetry algebra of a spin-$G$ structure
  16. ...and 5 more sections

Key Result

Proposition 3

A spin-$G$ structure $\widehat{\varpi}:\widehat{P}\to F_{SO}$ is reducible with lift $\psi:\widetilde{P}\to \widehat{P}$ if and only if there exists a spin structure $\varpi:P\to F_{SO}$, a lift $\varpi_Q:Q\to \overline{Q}$ of the structure group of the canonical $\overline{G}$-bundle along $p_G:G\t

Theorems & Definitions (22)

  • Definition 1: Spin-$G$ structure
  • Definition 2: Reducible spin-$G$ structure
  • Proposition 3
  • proof
  • Lemma 4
  • Definition 5
  • Lemma 6
  • proof
  • Proposition 7
  • Definition 8: Covariant Lie derivative
  • ...and 12 more